I asked Martin Davis to clarify his question of yesterday evening. Here
is what he answered [quotation necessary]. My response is below.
On Mon, 26 Jan 1998, Martin Davis wrote:
> >Martin,
> > I can't tell if you're missing something until I understand what your
> >question is. I must be missing something. Perhaps you can elaborate
> >before I try to answer publicly.
> >
> >Best,
> >Sol
> >
> Dear Sol,
>> I'm sorry I was obscure. If it hadn't been late at night, I probably
> wouldn't have had the huibris to write about a technical paper I hadn't read.
>> Here's what's bothering me:
>> Let A be some theorem about, say ODE's, that physicists use. Quantifiers
> over reals of course occur in A. You refer to a formal system T with
> variables of higher type in which A can be naturally translated into say a
> sentence *A, and *A is provable in T. Moreover T is a conservative extension
> of PA: no arithmetic sentence is provable in T not already provable in PA.
>> Now what I don't understand is how all of THIS is in any way germane to your
> assertion: "So the indispensability of that mathematics to science can't be
> counted as an argument for the real numbers somehow being embedded in the
> world."
>> Best, Martin
>> P.S. Please feel free to quote any of this on fom.
>>
It is germane to one way of formulating the indispensability arguments due
to Quine and Putnam. This was stated as follows by Maddy in the 1992
paper referred to in my posting:
"We have good reason to believe our best scientific theories, and
mathematical entities are indispensable to those theories, so we have good
reason to believe in mathematical entities. Mathematics is thus on an
ontological par with natural science. Furthermore, the evidence that
confirms scientific theories also confirms the required mathematics, so
mathematics and science are on an epistemological par as well." (Maddy
1992, p.78)
I take it that being on an "ontological par" with natural science means
somehow being part of the natural world. If not, what would it mean?
Perhaps there are other ways of formulating the indispensability arguments
that are supposed to count for some kind of justification of mathematics
without having this kind of conclusion. Perhaps Quine himself can be read
in a different way, at least in the following passage:
"So much of mathematics as is wanted for use in empirical science is for
me on a par with the rest of science. Transfinite ramifications are on
the same footing insofar as they come of a simplificatory rounding out,
but anything further is on a par with uninterpreted systems."
(Quine, review of Parsons' book, J. Philosophy 81 (1984),788.)
Here, "on a par with" is somewhat vague, in both instances. But Quine's
holism in general makes it hard to see how he would separate the status of
mathematical entities from physical entities. Quine experts may be able
to spell this out for us, if so.
I guess I have to add one more thing. What the conservation result in
question shows is that only very weak closure conditions on the real
numbers are needed for the various results in analysis that are derived in
the extension. Thus they do not support assumption that the "full set of
real numbers" in their usual set-theoretical conception is actually out
there. But then, what is out there of this character, if anything?
--Sol Feferman
PS. (Correction to my posting on foundations of naive category theory). It
has been pointed out to me that the correct locution is: "I don't chew my
cabbage twice". And, in fact, that's false.